The most common and most likely answer to this crossword clue is the 8 letter word PROPERLY
I've seen this crossword clue in The Guardian
P
R
O
P
E
R
L
Y
Buddy explains!

The straight part
We know that "Rightly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rightly = properly.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does PROPERLY mean?
PROPERLY adverb- In a proper manner, appropriately, suitably; correctly, justifiably
- Entirely; extremely; thoroughly.
- Individually; in one's own manner